Products of Languages with Counter

It is well known that varieties of rational languages are in one-to-one correspondence with varieties of finite monoids. This correspondence ofte., extends to operations on languages and on monoidc. We investigate the special case of the product of languages with counter, and describe the associated operations on monoids and varieties. R&emi. On sait que les varietes de IangagEs rationnels sont en correspondance bijective avec les varietes de monoi’des finis. Cette correspondance s’etend a de nombreuses operations sur les langages d’une part, sur ies mono’id es d’zutre part. Nous Ctudions le cas particu!ier du produit de langages avec compteur, et nous decrivons I’operation associee, sur les monoi’des et sur les varietes. The theory of form al languages is one of the bases of theoretical computer science. A central problem of this theory has been, since the origins in the 19X?s, the classificpfior? of rational languages. A very important tool for this task is the use of the syntactk monoid M(L) of a rational language L on an alphabet A. Indeed, many combinatorial properties of L correspond to a!gebraic properties of M(L). Eilenberg systematized this correspondence, and showed in 1975 that there is a one-to-one relation between certain families of finite monoids, called -varieties, and certak families of rational languages, called *-varieties. Numercus instances of this correspondence have been studied in detail since, such as the algebraic characterization of piecewise testable languages [14], locally testable languages [3,7] or, conversely, the co languages whose syntactic monoid is ,%-trivial [4, tains only solvable groups [ 151. hese ideas, the correspondent certain operations on monoids or in particular the relationship between concatenatio 0304-3975/90/$3.50 @ 1990, Elsevier Science Publishers B.V. (North-